MATH 270A California State Rational Numbers and Positive Integrers Questions

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prove or disprove following statements, explaining your reasoning. prove or disprove questions : total 7 questions

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4. (5 points) PROVE or DISPROVE the following statement.
There exist distinct rational numbers a and b such that (a – 1)(b − 1) = 1
5. (5 points) Suppose that A = {*,2,Ø} and B = {{0},{1,2}}. Let f : A + B be the function defined
by f(*) = {@}, f(2) = {0}, and f(0) = {1,2}. Determine whether f is one-to-one, onto, both, or
neither. Explain your answer.
PART III: PROOFS
Instructions: Prove the following statements, explaining your reasoning. Partial credit will be awarded for
an incorrect proof that shows progress towards a correct proof.
You must complete problem 1, and three of problems 2-5. That is, you will skip one of the problems 2-5.
Draw a large X through the problem you do not wish to be graded.
1. (20 points) Using mathematical induction, prove that for all positive integers n,
11 +19+27+ … + (8n +3) = n(4n+7)
2. (10 points) Prove: For all sets A, B and C,
A-(BUC) = (A – B) n(A-C).
3. (10 points) Suppose a and b are integers. Prove that if a +b is even, then a – b is also even.
4. (10 points) Prove that there exists no positive integer n such that 2n < n2 Purchase answer to see full attachment Tags: Mathematical induction rational numbers Division Algorithm Induction hypothesis Positive integrers User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.